\magnification = 2000 
\input amstex
\documentstyle{amsppt}
\input OurATOMacros
\input OurPlainGraphicsMacros


\Title {Pinkall's Flat Tori in $\Bbb S^3$}.

\section{Formulas}
 We can parametrize $\Bbb S^3$, considered as a submanifold of $\Bbb C^2$, by:
$$ F(u,\alpha,v) = (\cos(\alpha) e^{iu} e^{iv} , \sin(\alpha) e^{iu} e^{-iv}),$$
 where  $u \in [0,2 \pi)$, $\alpha \in [0, \pi/2]$, and $v \in [0, \pi]$.  
We will get the Pinkall Tori first as flat tori in $\Bbb S^3$ by taking $\alpha$ to be a 
 function of $v$, $\alpha := aa + bb \sin(\hbox{ee}\, 2v)$. 
 (The theory allows more general choices.)

\ni
 Next we stereographically project $\Bbb S^3$ from \lf
\clb {$p = (\cos(cc\cdot\pi),0,\sin(cc \cdot\pi),0)$}
 to get {\bf conformal} images of the flat  tori in $\Bbb S^3$. The lines $v = \hbox{const}$ are
 circles, the stereographic images of the Hopf circles $u\mapsto F(u,\alpha,v)$.  

\ni
Finally, by morphing  $0 \le \ff \le 2\pi$,
 we can isome\-trically rotate  $\Bbb S^3$ so that
the Hopf circle $v=0$ is the rotation axis. The stereographic image of this rotation
is a conformal transformation of $\Bbb R^3 \cup \{\infty\}$ which ``rotates'' $\Bbb R^3$
around a circle on the pictured torus. In the case $aa =\pi/4$ we obtain for $\ff=0$ and
$\ff=\pi$ the same torus, but inside and outside interchanged. This is best viewed
with the default `Two Sided User Coloration'. It can be selected from a submenu of the
Action Menu.

\section{Background and Explanations}
The tori which we usually see are, from the point of view
of complex analysis, rectangular tori. This means: They have
an orientation reversing symmetry and the set of fixed points of this
symmetry has two components. The well known tori of revolution have
isometric reflections with {\bf two} circles as fixed point sets. Of
course one tries to deform these tori to obtain nonrectangular ones.
Obviously one can destroy the mirror symmetry, but this does not imply
that one gets tori with a nonrectangular complex structure. The first
proof, by Garcia, that one can embed all tori in $\Bbb R^3$ was
nonconstructive and difficult. Pinkall's construction gives completely
explicit flat tori in $\Bbb S^3$. They have a one parameter family of
nonintersecting great circles on them which are parallel in the flat
geometry of the torus. Stereographic projection from  $\Bbb S^3$ to
 $\Bbb R^3$ gives conformal images of these flat tori.

\ni
   The great circles $u \mapsto F(u,\alpha,v)$ are the orbits of the Hopf-action
 of $\Bbb S^1$ on $\Bbb S^3$, $(u,p)\mapsto e^{iu} p$. Each such ``Hopf Fiber'' 
lies in one of the parallel tori  $\alpha = $ constant, and the great circles 
$\alpha \mapsto F(u,\alpha,v)$, meet these tori orthogonally, 
so that $\alpha$ measures the dsitance between them. 
Taking $\alpha = \pi/4$ gives the ``Clifford Torus'' in $\Bbb S^3$, a minimal 
embedding of the square torus. For all $\alpha$ in $(0,\pi/4)$, the lengths of 
two orthogonal generators of the corresponding torus are $2 \pi \cos(\alpha) $
and $2 \pi \sin(\alpha)$.

\bigskip \bigskip\goodbreak
\ni
   The set of Hopf-Fibers with the natural distance in $\Bbb S^3$ is a metric space,
isometric to the 2-sphere of radius $1/2$ (curvature $4$). In fact, $(\alpha,2v)$ 
are polar coordinates for this sphere.  Pinkall observed that for any closed curve
$(\alpha(s),\phi(s))$ on this sphere (with $\alpha(s)$ never equal to $0$ or $\pi/2$)
$(u,s) \mapsto (F(u,\alpha(s),v(s))$ gives an immersed torus. On these tori we 
still have the Hopf-Fibers, and since these are equidistant it follows that
the metric is flat. Pinkall proved that the length and area of the curve in 
$\Bbb S^2$ determine the conformal structure of the torus, and that all conformal 
structures occur.
 
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Observe that the usual tori of revolution in $\Bbb R^3$ are rectangular, and
most of the Pinkall tori shown by 3D-XplorMath are very different from these.
The tori with aa $= \pi/4$ are all rhombic, because they can be rotated into themselves 
by $180^\circ$ rotations (in $\Bbb S^3$, not $\Bbb R^3$) around any of the Hopf-Fibers
on them. A cyclic morph with $ 0 \le \ff  \le 2\pi$ rotates around the circle $v=0$. This
rotation is an anti-involution of the torus with the circle as the (connected) fixed point 
set---only rhombic tori have such anti-involutions. (The square torus is rectangular and rhombic.)
 
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The program takes $\alpha(v) := aa + bb \sin(\hbox{ee}\, 2v)$ (with $ee= 3$ for the default
image and $ee= 5$ for the default morph). These tori are
shown in $\Bbb R^3$ by using the (conformal) stereographic projection of 
$\Bbb S^3\setminus\{p\} \to \Bbb R^3$, where  the projection point $p$ is given by 
{$p := (\cos(cc\cdot\pi),0,\sin(cc \cdot\pi),0).$} 
Morphing $cc$ also gives conformal deformations of these tori.

\ni
H.K.
 




\bye
